## Algebra - Matrices

### Chapter 1. Introduction to matrices

• Definition 1.1: A Matrix is an ordered rectangular array of numbers or functions arranged in rows and columns. The numbers or functions are called the elements or entries of the matrix.

• Definition 1.2: A matrix having $m$ rows and $n$ columns is called a matrix of order $m \times n$ where $m$ and $n$ are positive integers greater than 0.

• Notations: Matrices are generally denoted by capital letters: $A$,$B$,$C$, etc.
The elements of Matrices are generally denoted by small letters: $a$,$b$,$c$, etc.

• We denote $A=[a_{ij}]_{m \times n}$ to indicate that $A$ is a matrix of order $m \times n$.
In general an $m \times n$ matrix is of the form $\begin{bmatrix} a_{11} & a_{12} & a_{13} & \cdots& a_{1j} & \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23} & \cdots& a_{2j} & \cdots & a_{2n} \\ \vdots &\vdots&\vdots&\cdots&\vdots&\cdots&\vdots \\ a_{i1} & a_{i2} & a_{i3} & \cdots& a_{ij} & \cdots & a_{in} \\ \vdots &\vdots&\vdots&\cdots&\vdots&\cdots&\vdots \\ a_{m1} & a_{m2} & a_{m3} & \cdots& a_{mj} & \cdots & a_{mn} \\ \end{bmatrix} _{m \times n}$
In particular a $2 \times 2$ matrix is of the form $\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} _{2 \times 2}$
In particular a $3 \times 3$ matrix is of the form $\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} _{3 \times 3}$

• Definition 1.3: A matrix with real elements is called a real matrix.
Example: $\begin{bmatrix} 1 & -5 & \sqrt{5} \\ 3 & 5 & 4 \end{bmatrix}$

• Definition 1.4: A matrix with complex elements is called a complex matrix.
Example: $\begin{bmatrix} i & -5+4i & 0 \\ 3+i & -i & 4 \\ 0 & 6i & -1 \end{bmatrix}$

• Remark: Every real matrix is a complex matrix but converse may not be true.

• Definition 1.5: Any $m \times n$ matrix, where $m \neq n$ is called a rectangular matrix.
Example: $\begin{bmatrix} 1&2&5&-7\\ 9&10&-8&1\\ 10&12&-5&7\\ \end{bmatrix}$

• Definition 1.6: Any $m \times m$ (or $n \times n$) matrix, is called a square matrix.
Example: $\begin{bmatrix} 1&2&5&-7\\ 9&10&-8&1\\ 10&12&-5&7\\ 1&2&3&4 \end{bmatrix}$

• Definition 1.7: Any $1 \times m$ (or $1 \times n$) matrix, is called a row matrix.
Example: $\begin{bmatrix} 1&2&3&4 \end{bmatrix}$

• Definition 1.8: Any $m \times 1$ (or $n \times 1$) matrix, is called a column matrix.
Example: $\begin{bmatrix} 1\\ 2\\ 3\\ \end{bmatrix}$

• Defintion 1.9: An element $a_{ij}$ of a square matrix $A=[a_{ij}]$ of order $n$ (i.e., of order $n \times n$) is said to be a diagonal element if $i=j$.
Example: $A= \begin{bmatrix} 1&4&6\\ 2&5&7\\ 3&6&8\\ \end{bmatrix}$
Diagonal elements of $A$ are $1, 5$ and $8$

• Definition 1.10: If $A=[a_{ij}]$ is a square matrix, then the sum of its diagonal elements is called the trace of the matrix $A$.
Example: $A= \begin{bmatrix} 6&2&3\\ 4&0&6\\ 7&8&-3\\ \end{bmatrix}$
Diagonal elements of $A$ are $6, 0$ and $-3$
Trace of $A$ is $3$

• Definition 1.11: A square matrix $A=[a_{ij}]$ is called diagonal matrix if $a_{ij}=0$ when $i \neq j$.
Example: $\begin{bmatrix} 1&0&0\\ 0&5&0\\ 0&0&-6\\ \end{bmatrix}$

• Definition 1.12: A diagonal matrix $A=[a_{ij}]$ is called scalar matrix if $a_{ij}=0$ when $i \neq j$ and $a_{ij}=k$ when $i=j$ where $k$ is a fixed element.
Example: $\begin{bmatrix} 5&0&0\\ 0&5&0\\ 0&0&5\\ \end{bmatrix}$

• Definition 1.13: A scalar matrix $A=[a_{ij}]$ is called identity matrix if $a_{ij}=0$ when $i \neq j$ and $a_{ij}=1$ when $i=j$
Example 1: Identity matrix of order $2 \times 2$ $\begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}$
Example 2: Identity matrix of order $3 \times 3$ $\begin{bmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{bmatrix}$
Notation: Identity matrix is denoted by $I$. Identity matrix of order $n$ is denoted by $I_n$

• Definition 1.14: If every element above or below the diagonal is zero, then the matrix is called triangular matrix.
In particular, if every element above the diagonal is zero, then the matrix is called lower triangular matrix.
Example: $\begin{bmatrix} 1&0&0\\ 5&2&0\\ 4&6&3\\ \end{bmatrix}$
In particular, if every element below the diagonal is zero, then the matrix is called upper triangular matrix.
Example: $\begin{bmatrix} 1&6&8\\ 0&2&6\\ 0&0&3\\ \end{bmatrix}$
Remark: A Diagonal matrix is a square matrix which is both lower and upper triangular matrix.

• Definition 1.15: A matrix in which every element is zero is called zero matrix or null matrix.
Example 1: zero matrix of order $3 \times 3$ $\begin{bmatrix} 0&0&0\\ 0&0&0\\ 0&0&0\\ \end{bmatrix}$
Example 2: zero matrix of order $2 \times 3$ $\begin{bmatrix} 0&0&0\\ 0&0&0\\ \end{bmatrix}$
Notation: Zero matrix is denoted by $O$.
Remark: Zero matrix may be square or rectangular.

• Definition 1.16: Determinant is a value/quantity obtained by the addition of products of the elements of a square matrix according to a given rule.
Notation: The determinant of a matrix $A$ is denoted det$(A)$ or $|A|$.
Let $A= \begin{bmatrix} a_{11}&a_{12}\\ a_{21}&a_{22} \end{bmatrix} _{2 \times 2}$
then $|A|=a_{11}a_{22}-a_{12}a_{21}$
Let $A= \begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{bmatrix} _{3 \times 3}$
then $|A|=a_{11} \begin{vmatrix} a_{22}&a_{23}\\ a_{32}&a_{33} \end{vmatrix} - a_{12}\begin{vmatrix} a_{21}&a_{23}\\ a_{31}&a_{33} \end{vmatrix} + a_{31} \begin{vmatrix} a_{21}&a_{22}\\ a_{31}&a_{32} \end{vmatrix}$

• Definition 1.17: A square matrix $A$ is said to be $\textbf{singular matrix}$ if $|A|=0$.
Example: Let $A= \begin{bmatrix} 2&4\\ 4&8 \end{bmatrix} _{2 \times 2}$ then $|A|= \begin{vmatrix} 2&4\\ 4&8 \end{vmatrix} =2 \times 8 - 4 \times 4 = 16 - 16 = 0$
Hence $A$ is singular matrix.

• Definition 1.18: A square matrix $A$ is said to be $\textbf{non-singular matrix}$ if $|A|\neq 0$.
Example: Let $A= \begin{bmatrix} 2&4\\ 5&2 \end{bmatrix} _{2 \times 2}$ then $|A|= \begin{vmatrix} 2&4\\ 5&2 \end{vmatrix} =2 \times 2 - 4 \times 5 = 4 - 20 = -16 \neq 0$
Hence $A$ is non-singular matrix.

• Definition 1.19: If $A=[a_{ij}]_{m \times n}$ is a matrix, then the matrix obtained by interchanging rows to columns of $A$ is called $\textbf{transpose of a matrix}$ $A$.
Notation: Transpose of a matrix $A$ is denoted by $A'$ or $A^T$
If $A=[a_{ij}]_{m \times n}$ then $A'=[a_{ji}]_{n \times m}$
Example: $A= \begin{bmatrix} 1&2&3\\ 2&4&5 \end{bmatrix} _{2 \times 3}$ then $A'= \begin{bmatrix} 1&2\\ 2&4\\ 3&5 \end{bmatrix} _{3 \times 2}$

• Definition 1.20: If $A$ is a matrix, then the matrix obtained by replacing its elements by corresponding conjugates (complex or real) is called $\textbf{conjugate of the matrix}$ $A$.
Notation: Conjugate matrix of $A$ is denoted by $\bar{A}$
Example 1: Let $A= \begin{bmatrix} 1&1&-2\\ 0&-2&4\\ -2&9&-1 \end{bmatrix}$ then $\bar{A}= \begin{bmatrix} 1&1&-2\\ 0&-2&4\\ -2&9&-1 \end{bmatrix}$
Example 2: Let $A= \begin{bmatrix} 1+i&1&i\\ 0&2&-2+i\\ -2-i&9-i&-1 \end{bmatrix}$ then $\bar{A}= \begin{bmatrix} 1-i&1&-i\\ 0&2&-2-i\\ -2+i&9+i&-1 \end{bmatrix}$

• Two matrices $A=[a_{ij}]$ and $B=[b_{ij}]$ are said to be equal if
(i) $A$ and $B$ are of same order.
(ii) $a_{ij}$$=$$b_{ij}$ for all values of $i$ and $j$.

• Addition of matrices: If $A=[a_{ij}]$ and $B=[b_{ij}]$ are two matrices of same order $m \times n$, then their sum $A+B$ is defined to be the matrix of same order $m \times n$ obtained by adding the corresponding elements of $A$ and $B$.
i.e., If $A=[a_{ij}]_{m \times n}$ and $B=[b_{ij}]_{m \times n}$, then $A+B$$=$$[a_{ij}+b_{ij}]_{m \times n}$
Properties of matrix addition:
Matrix addition is commutative: If $A$ and $B$ are any two matrices of order $m \times n$, then $A+B=B+A$.
Matrix addition is associative: If $A$,$B$ and $C$ are any matrices of order $m \times n$, then $A+(B+C)=(A+B)+C$.
Existence of additive identity: If $A$ is any matrix of order $m \times n$, then $\exists$ a null matrix $O$ of same order, such that $A+O=O+A=A$.
Existence of additive inverse: If $A$ is any matrix of order $m \times n$, then $\exists$ a matrix $-A$ ($-A$ is called the additive inverse of $A$), such that $A+(-A)=(-A)+A=O$.

• Difference of matrices: If $A=[a_{ij}]$ and $B=[b_{ij}]$ are two matrices of same order $m \times n$, then their difference $A-B$ is defined to be the matrix of same order $m \times n$ obtained by subtracting the corresponding elements of $A$ from $B$.
i.e., If $A=[a_{ij}]_{m \times n}$ and $B=[b_{ij}]_{m \times n}$, then $A-B$$=$$[a_{ij}-b_{ij}]_{m \times n}$

• Multiplication of a matrix by a scalar: If $A=[a_{ij}]$ is matrix of order $m \times n$ and $k$ is any scalar, then $k A$ is a matrix of same order $m \times n$ obtained by multiplying every element of $A$ by $k$.
i.e., If $A=[a_{ij}]_{m \times n}$ and $k$ is any scalar, then $k A$$=$$[k a_{ij}]_{m \times n}$

• Multiplication of matrices: If $A=[a_{ij}]_{m \times n}$ and $B=[b_{ij}]_{n \times p}$ are two matrices such that the number of columns in $A$ is equal to the number of rows in $B$, then the product of $A$ and $B$ is defined to be the matrix $C=[c_{ij}]_{m \times p}$ where $c_{ik}=\sum\limits_{j=1}^{n}a_{ij}b_{jk}$, where $i=1,2,...,m$ and $k=1,2,....p$.
Properties of matrix multiplication:
Matrix multiplication is associative: If $A$,$B$ and $C$ are any matrices, then $A(BC)=(AB)C$
Existence of multiplicative identity: If $A$ is any matrix, then $\exists$ a identity matrix $I$, such that $AI=IA=A$
Existence of multiplicative inverse: If $A$ is any non singular matrix, then $\exists$ a matrix $A^{-1}$ ($A^{-1}$ is called the inverse of $A$), such that $AA^{-1}=A^{-1}A=I$

• Element wise multiplication of matrices: If $A=[a_{ij}]$ and $B=[b_{ij}]$ are two matrices of same order $m \times n$, then their element wise multiplication $A.B$ is defined to be the matrix of same order $m \times n$ obtained by multiplying the corresponding elements of $A$ and $B$.
i.e., If $A=[a_{ij}]_{m \times n}$ and $B=[b_{ij}]_{m \times n}$, then $A.B$$=$$[a_{ij}b_{ij}]_{m \times n}$

• Definition 1.21: A square matrix is said to be symmetric if $A=A'$.
Example: $A= \begin{bmatrix} 1&1/2&3\\ 1/2&2&5\\ 3&5&-2 \end{bmatrix}$ then $A'= \begin{bmatrix} 1&1/2&3\\ 1/2&2&5\\ 3&5&-2 \end{bmatrix}$
Thus $A=A'$. Hence $A$ is a symmetric matrix.

• Definition 1.22: A square matrix is said to be skew-symmetricif $A=-A'$.
Example: $A= \begin{bmatrix} 0&-1&3.5\\ 1&0&6\\ -3.5&-6&0 \end{bmatrix}$ then $A'= \begin{bmatrix} 0&1&-3.5\\ -1&0&-6\\ 3.5&6&0 \end{bmatrix}$ $= \begin{bmatrix} 0&-1&3.5\\ 1&0&6\\ -3.5&-6&0 \end{bmatrix} =-A$
Thus $A'=-A$. Hence $A$ is a skew-symmetric matrix.

• Every square matrix can be expressed uniquely as a sum of symmetric and skew-symmetric matrix. Let $A$ be any square matrix then $\frac{A+A'}{2}$ is symmetric and $\frac{A-A'}{2}$ is skew symmetric. Thus $A=\frac{A+A'}{2}+\frac{A-A'}{2}$

• Definition 1.23: A square matrix is said to be Hermitian if $A=(\bar{A})'$, where $\bar{A}$ is the conjugate matrix of $A$.
Example: $A= \begin{bmatrix} 1&2-3i&4+5i\\ 2+3i&0&5-6i\\ 4-5i&5+6i&2 \end{bmatrix}$ $\Rightarrow$ $\bar{A}= \begin{bmatrix} 1&2+3i&4-5i\\ 2-3i&0&5+6i\\ 4+5i&5+6i&2 \end{bmatrix} \Rightarrow$ $(\bar{A})'= \begin{bmatrix} 1&2-3i&4+5i\\ 2+3i&0&5+6i\\ 4-5i&5-6i&2 \end{bmatrix}$
Thus $A=(\bar{A})'$. Hence $A$ is a Hermitian matrix.

• Definition 1.24: A square matrix is said to be skew-Hermitian if $A=-(\bar{A})'$, where $\bar{A}$ is the conjugate matrix of $A$.
Example: $A= \begin{bmatrix} i&1-i&2\\ -1-i&3i&i\\ -2&i&0 \end{bmatrix}$ $\Rightarrow$ $\bar{A}= \begin{bmatrix} -i&1+i&2\\ -1+i&-3i&-i\\ -2&-i&0 \end{bmatrix}$ $\Rightarrow$ $(\bar{A})'= \begin{bmatrix} -i&-1+i&-2\\ 1+i&-3i&-i\\ 2&-i&0 \end{bmatrix}$ $=- \begin{bmatrix} i&1-i&2\\ -1-i&3i&i\\ -2&i&0 \end{bmatrix} =-A$
Thus $(\bar{A})'=-A$. Hence $A$ is a skew-Hermitian matrix.

• Every symmetric matrix is Hermitian and every skew-symmetric matrix is skew-Hermitian but converse may not be true.

• Definition 1.25: Transpose of the conjugate of a matrix $A$ is called transposed conjugate or tranjugate of matrix $A$.
Notation: Transposed conjugate of a matrix A is denoted by $A^\Theta$ i.e., $A^\Theta=(\bar{A})'$

• Definition 1.26: A square matrix is said to be orthogonal if $A'A=I$, where $A'$ is the transpose of a matrix $A$.

• Definition 1.27: A square matrix is said to be unitary if $A^\Theta A =I$, where $A^\Theta$ is the transpose of conjugate of a matrix $A$.

• Definition 1.28: Minor of an element $a_{ij}$ is the determinant obtained by deleting its $i^{th}$ row and $j^{th}$ column in which $a_{ij}$ lies.
Notation: Minor of $a_{ij}$ is denoted by $M_{ij}$.

• Definition 1.29: Cofactor of an element $a_{ij}$ is defined as $(-1)^{i+j}M_{ij}$.
Notation: Cofactor of $a_{ij}$ is denoted by $A_{ij}$.

• Definition 1.30: If $A=[a_{ij}]$ is a square matrix, then adjoint of $A$ is defined as the transpose of the matrix $[A_{ij}]$, where $[A_{ij}]$ is the cofactor of the element $a_{ij}$.
Notation: Adjoint of $A$ is denoted by adj$A$.
Example: Let $A= \begin{bmatrix} 1&4&6\\ 0&9&8\\ -1&2&3 \end{bmatrix}$
Minor matrix of $A$ is $\begin{bmatrix} 11&8&9\\ 0&9&6\\ -22&8&9 \end{bmatrix}$
Co factor matrix of $A$ is $\begin{bmatrix} 11&-8&9\\ 0&9&-6\\ -22&-8&9 \end{bmatrix}$
adj$A=\begin{bmatrix} 11&-8&9\\ 0&9&-6\\ -22&-8&9 \end{bmatrix}'$ $=\begin{bmatrix} 11&0&-22\\ -8&9&-8\\ 9&-6&9 \end{bmatrix}$

• Exercise 1.1: Classify the following matrices:
(1.) $\begin{bmatrix} 1&3&5\\8&5&-1 \end{bmatrix}$
(2.) $\begin{bmatrix}-1&3&5\\0&12&3\\8&5&-1 \end{bmatrix}$
(3.) $\begin{bmatrix} 1&0\\0&1 \end{bmatrix}$
(4.) $\begin{bmatrix} -3&0&0\\0&-3&0\\0&0&-3 \end{bmatrix}$
(5.) $\begin{bmatrix} 1&3&5&2 \end{bmatrix}$
(6.) $\begin{bmatrix} 1&3&5\\ 0&5&-1\\ 0&0&12 \end{bmatrix}$

• Exercise 1.2: Are the following matrices equal?
(1.)$\begin{bmatrix} 10&23&0\\ -1&4&\sqrt{6}\\ 4&-8&12 \end{bmatrix}$, $\begin{bmatrix} 10&23&0\\ 1&4&-\sqrt{6}\\ 4&-8&-12 \end{bmatrix}$
(2.)$\begin{bmatrix} 1&2&0\\ 6&7&8 \end{bmatrix}$, $\begin{bmatrix} 1&2&0\\ 6&7&8 \end{bmatrix}$
(3.)$\begin{bmatrix} 1&1/2&3\\ 5&3-4i&7\\ 6&2&i\\ 1&0&1-i \end{bmatrix}$, $\begin{bmatrix} 1&1/2&3\\ 5&3+4i&7\\ 6&2&-i\\ 1&0&1+i \end{bmatrix}$

• Exercise 1.3: Find $x$ and $y$, if
(1.) $2 \begin{bmatrix} 1&3\\ 0&x \end{bmatrix}$+ $\begin{bmatrix} y&0\\ 1&2 \end{bmatrix} = \begin{bmatrix} 5&6\\ 1&8 \end{bmatrix}$
(2.) $x \begin{bmatrix} 2\\ 3 \end{bmatrix}$+ $y \begin{bmatrix} -1\\ 1 \end{bmatrix} = \begin{bmatrix} 10\\ 5 \end{bmatrix}$

• Exercise 1.4: Find $X$ and $Y$, if
(1.) $X+Y=$ $\begin{bmatrix} 7&0\\ 2&5 \end{bmatrix}$ and $X-Y=$ $3 \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}$
(2.) $X+Y$ $=\begin{bmatrix} 5&2\\ 0&9 \end{bmatrix}$ and $X-Y=$ $\begin{bmatrix} 3&6\\ 0&-1 \end{bmatrix}$

• Exercise 1.5: Given $A= \begin{bmatrix} 2&3&5\\ 5&4&2\\ 2&5&9 \end{bmatrix}$ and $B= \begin{bmatrix} 5&-9&6\\ 2&3&-5\\ 4&-9&7 \end{bmatrix}$ Compute $A+B$, $A-B$, $AB$, $BA$, $3A-B$

• Exercise 1.6: Find the product of the following matrices
(1.) $\begin{bmatrix} a&b\\ -b&a \end{bmatrix}$ , $\begin{bmatrix} a&-b\\ b&a \end{bmatrix}$
(2.) $\begin{bmatrix} 1\\ 2\\ 3 \end{bmatrix}$ , $\begin{bmatrix} 4&5&6 \end{bmatrix}$
(3.) $\begin{bmatrix} 1 &-2\\ 2&3 \end{bmatrix}$ , $\begin{bmatrix} 1&2&3\\ 2&3&1 \end{bmatrix}$
(4.) $\begin{bmatrix} 2&3&4\\ 3&4&5\\ 4&5&6 \end{bmatrix}$ , $\begin{bmatrix} 1&-3&5\\0&2&4\\3&0&5\end{bmatrix}$
(5.) $\begin{bmatrix} 2&1\\ 3&2\\ -1&1 \end{bmatrix}$ , $\begin{bmatrix} 1&0&1\\ -1&2&1 \end{bmatrix}$
(6.) $\begin{bmatrix} 3&-1&3\\ -1&0&2 \end{bmatrix}$ , $\begin{bmatrix} 2&-3\\2&0\\3&1 \end{bmatrix}$

• Exercise 1.7: If $A=\begin{bmatrix} 1&0&2\\ 0&2&1\\ 2&0&3 \end{bmatrix}$ Find $A^2$ and $A^3$

• Exercise 1.8: If $A'=\begin{bmatrix} 3&4\\ -1&2\\ 0&1 \end{bmatrix}$ and $B=\begin{bmatrix} -1&2&1\\ 1&2&3 \end{bmatrix}$, then verify that
(1.) $(A+B)'=A'+B'$
(2.) $(A-B)'=A'-B'$
(3.) $(AB)'=B'A'$

• Exercise 1.9: Show that the matrix $A= \begin{bmatrix} 1&-1&5\\ -1&2&1\\ 5&1&3 \end{bmatrix}$ is a symmetric matrix.

• Exercise 1.10: Show that the matrix $A= \begin{bmatrix} 0&1&-2\\ -1&0&5\\ 2&-5&0 \end{bmatrix}$ is a skew-symmetric matrix.

• Exercise 1.11: For a matrix $A=\begin{bmatrix} 1&5\\ 6&7 \end{bmatrix}$, Show that $A+A'$ is a symmetric matrix and $A-A'$ is a skew symmetric matrix.

• Exercise 1.12: Express the following matrices as the sum of symmetric and skew symmetric matrix:
(1.) $\begin{bmatrix} 8&4&-3\\ 2&1&1\\ 1&2&1 \end{bmatrix}$
(2.) $\begin{bmatrix} 2&5&7\\ 3&1&2\\ 9&-4&8 \end{bmatrix}$
(3.) $\begin{bmatrix} 1&2&2\\ 2&0&1\\ 0&4&-3 \end{bmatrix}$
(4.) $\begin{bmatrix} 5&4&-1\\ 2&0&3\\ -1&2&1 \end{bmatrix}$

• Exercise 1.13: Show that the matrix $\begin{bmatrix} 1&2+3i&-1\\2-3i&\sqrt{5}&\sqrt{2}+i\\-1&\sqrt{2}-i&3/2\end{bmatrix}$ is Hermitian.

• Exercise 1.14: Show that the matrix $\begin{bmatrix} -i&1+2i&2-3i\\-1+2i&2i&\sqrt{2}-i\\-2-3i&-\sqrt{2}-i&0 \end{bmatrix}$ is skew-Hermitian.

• Exercise 1.15: Show that the matrix $\begin{bmatrix} 0&2+i&i\\ -2+i&0&1-i\\ i&-1-i&0 \end{bmatrix}$ is skew-Hermitian.

• Exercise 1.16: Evaluate the determinants:
(1.) $\begin{vmatrix} 2&4\\ -5&-1 \end{vmatrix}$
(2.) $\begin{vmatrix} 3 &-1& -2\\1&1&-2\\2&3&1 \end{vmatrix}$

• Exercise 1.17: Find the cofactors of the elements of the following determinants:
(1.) $\begin{vmatrix} 2&-4\\0&2 \end{vmatrix}$
(2.) $\begin{vmatrix} 5 &3 &8\\1&0&1\\-2&-1&1 \end{vmatrix}$

• Exercise 1.18: Find the adj$A$ for the following matrices:
(1.) $A=\begin{bmatrix} -1&5\\-3&2 \end{bmatrix}$
(2.) $A=\begin{bmatrix} 2&-2\\4&3 \end{bmatrix}$
(3.) $A=\begin{bmatrix} 1&2&3\\0&2&4\\0&0&5 \end{bmatrix}$
(4.) $A=\begin{bmatrix} 1&-1&2\\3&0&-2\\1&0&3 \end{bmatrix}$
(5.) $A=\begin{bmatrix} 1&0&0\\3&3&0\\5&2&-1 \end{bmatrix}$
(6.) $A=\begin{bmatrix} 1&-1&2\\ 0&2&-3\\ 3&-2&4 \end{bmatrix}$
(7.) $A=\begin{bmatrix} 2&1&3\\4&-1&0\\-7&2&1 \end{bmatrix}$
(8.) $A=\begin{bmatrix} 1&0&0\\0&\cos\alpha&\sin\alpha\\0&\sin\alpha&-\cos\alpha \end{bmatrix}$

• Exercise 1.19: If $A= \begin{bmatrix} 2&3&4\\ 4&6&8\\ 1&2&5 \end{bmatrix}$, Show that $A($adj$A)=0$

• Exercise 1.20: If $A= \begin{bmatrix} 1&3&2\\ 2&1&3\\ 3&2&1 \end{bmatrix}$, Find $|$adj$A|$

• Exercise 1.21: If $A= \begin{bmatrix} 1&2\\3&4 \end{bmatrix}$, Show that $A($adj$A)=($adj$A)A=|A|I$

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